Properties

Label 2-3744-104.77-c1-0-63
Degree $2$
Conductor $3744$
Sign $-0.412 + 0.910i$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.68·5-s − 4.15i·7-s + 4.35·11-s + (−3.53 + 0.726i)13-s − 5.87·17-s − 5.71·19-s + 3.62·23-s + 2.23·25-s + 3.08i·29-s − 9.28i·31-s − 11.1i·35-s + 2.69·37-s − 11.1i·41-s − 3.80i·43-s − 4.91i·47-s + ⋯
L(s)  = 1  + 1.20·5-s − 1.56i·7-s + 1.31·11-s + (−0.979 + 0.201i)13-s − 1.42·17-s − 1.31·19-s + 0.756·23-s + 0.447·25-s + 0.573i·29-s − 1.66i·31-s − 1.88i·35-s + 0.443·37-s − 1.73i·41-s − 0.580i·43-s − 0.716i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.412 + 0.910i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1585, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ -0.412 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.799029689\)
\(L(\frac12)\) \(\approx\) \(1.799029689\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (3.53 - 0.726i)T \)
good5 \( 1 - 2.68T + 5T^{2} \)
7 \( 1 + 4.15iT - 7T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
17 \( 1 + 5.87T + 17T^{2} \)
19 \( 1 + 5.71T + 19T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 - 3.08iT - 29T^{2} \)
31 \( 1 + 9.28iT - 31T^{2} \)
37 \( 1 - 2.69T + 37T^{2} \)
41 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 + 3.80iT - 43T^{2} \)
47 \( 1 + 4.91iT - 47T^{2} \)
53 \( 1 + 1.17iT - 53T^{2} \)
59 \( 1 - 2.29T + 59T^{2} \)
61 \( 1 + 7.05iT - 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 2.08iT - 71T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 9.73T + 83T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 - 5.13iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.417276574168459992367998440385, −7.18849283112006149595960355125, −6.87208960862944202053175620584, −6.25031054762548125810645670261, −5.24859704062323917144294070862, −4.25199483288975713574124635381, −3.93562357978125545531880361307, −2.41756569679320555919249314405, −1.74436092226752312869185379652, −0.47983822295307819993214674586, 1.53583433469821506692681233970, 2.31111550387432801456233132082, 2.92679125792682767286712499709, 4.40213296513089773547079259360, 4.98363774165843043044534637369, 5.97565651828623537844267118102, 6.34280451998579018829092347408, 7.00748044806286195819007411970, 8.324187983660701567771315529641, 8.883077683389875645394762567414

Graph of the $Z$-function along the critical line